Performing a playwright's work for the first time is an experience quite unlike reading it, or seeing it staged. Now you're on the inside, feeling the author's emotions and tasting his language in a very personal way. With Arthur Miller, I have been astonished by the rawness of his writing, by his anger and humour. His play Broken Glass, which I have been performing for almost a year now, has the fearlessness of a young man's response to the world – yet Miller wrote it when he was 78.

The play is set in 1938. In the background, there is Kristallnacht in Germany; in the foreground, a Jewish community in Brooklyn react to the news. Sylvia Gellburg is traumatised, as if she knows this is only a curtain-raiser to the horrors to come. Meanwhile, her husband Phillip (my role) wants to ignore it; he is uneasy with his own Jewishness. And then there is Dr Hyman, who develops a relationship with Sylvia that crosses the boundaries between medic and patient. He has studied in Germany, and maintains that such a cultured people aren't capable of further brutality.

Miller's is an original view of the Holocaust, seen from the safety of New York. When he wrote it, in 1993, he was motivated by contemporary events in Rwanda and Serbia. Once again, he noted, atrocities were happening elsewhere, and we were just looking on helplessly. The play's director, Iqbal Khan, describes this as one of the main themes of the play: "The things we allow to happen – whether personally or politically".

When, a year ago, lqbal first asked me to play Phillip Gellburg, it was impossible not to notice his own ethnicity. Once I got to know him better, I asked the inevitable question: "What's a nice Muslim boy doing directing a nice Jewish play?" lqbal explained that he has no religious faith (in the same way that I am a purely secular Jew), but that he feels sensitive to the characters because of the experiences of his father's generation, Pakistanis who emigrated to Britain in the 1960s.

"The options were either to assimilate, and remove the sense that you were a 'threat' to your 'hosts'," he explained, "or to entrench yourself among your own people. Fear influenced every action. Fear of rejection, persecution, exclusion, or, conversely, of betraying your ancestral origins." In Broken Glass, "fear", "frightened" and "afraid" are the most frequently used words.

My own search for Gellburg benefited enormously from an identity crisis I had as a young man. I wasn't open about being gay, I kept stumm about being Jewish, and I was ashamed of being a white South African. I ended up in so many closets I didn't know which key was which. Those early attempts to flee from being Antony Sher were useful when it came to being Gellburg, a man uncomfortable in his own skin.

After an initial run of Broken Glass in London last year, I was asked to play Gellburg again, in a new South African production. The cast was local and we performed at a new venue in my birthplace, Cape Town: the Fugard theatre has a spectacular setting at the foot of Table Mountain. Its namesake, South Africa's greatest playwright, Athol Fugard, was rehearsing his latest play there while we were performing. I asked what it was like to have a theatre bearing his name. He laughed: "Agh, man, I'm too embarrassed to say it – I just call it 'that theatre in District Six'." (In 1966, the district's mixed race population were forced out of their homes and relocated to the barren Cape Flats.)

South Africa's past began to inform the play. Gellburg boasts that he is the only Jew in his firm, ignoring the fact that his boss displays the casual antisemitism of the period. These scenes reminded South Africans of life under apartheid, and the behaviour that a black man, even in a position of relative privilege, would be forced to adopt with his "baas" or "master". Especially potent, both to cast and audience, was the most challenging question Miller poses: what do you do, personally, when you witness evil in action?

In the South Africa of my youth, it was part of daily life. Very few people stood up and protested. I suspect that Miller, had he been South African, might have been one of them. His refusal to give evidence to McCarthy's House Un-American Activities Committee in 1956 proved he was a brave man.

Back in London, I embarked on a revival of last year's production. Any apprehension I might have felt about revisiting familiar territory has been dispelled by lqbal's passion for a particular aspect of Miller's writing: its unpredictability. As he said: "With Miller, we are constantly surprised by the possibilities in each of us, for cruelty or compassion or absurdity." Each new cast has forced me to be jolted by the newness of the events in the story.

While the casts have changed, one audience reaction has remained constant: they want to know why the play isn't better known. This could have something to do with the fact that it came in the latter half of Miller's career, when he was underrated in the US (though celebrated here). In 1994, after a lukewarm reception on Broadway, the play was a big success at the National theatre in London, winning an Olivier for best new play.